Anisotropy of wood vibrational properties: dependence on grain angle and review of literature data

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Anisotropy of wood vibrational properties: dependence

on grain angle and review of literature data

Iris Brémaud, Joseph Gril, Bernard Thibaut

To cite this version:

Iris Brémaud, Joseph Gril, Bernard Thibaut. Anisotropy of wood vibrational properties: dependence

on grain angle and review of literature data. Wood Science and Technology, Springer Verlag, 2011, 45

(4), pp.735-754. �10.1007/s00226-010-0393-8�. �hal-00804242�

Anisotropy

of wood vibrational properties: dependence

on

grain angle and review of literature data

Iris Bre´maud • Joseph Gril • Bernard Thibaut

Abstract The anisotropy of vibrational properties influences the acoustic

behav-iour of wooden pieces and their dependence on grain angle (GA). As most pieces of wood include some GA, either for technological reasons or due to grain deviations inside trunks, predicting its repercussions would be useful. This paper aims at evaluating the variability in the anisotropy of wood vibrational properties and analysing resulting trends as a function of orientation. GA dependence is described by a model based on transformation formulas applied to complex compliances, and literature data on anisotropic vibrational properties are reviewed. Ranges of vari-ability, as well as representative sets of viscoelastic anisotropic parameters, are defined for mean hardwoods and softwoods and for contrasted wood types. GA-dependence calculations are in close agreement with published experimental results and allow comparing the sensitivity of different woods to GA. Calculated trends in

damping coefficient (tand) and in specific modulus of elasticity (E0/q) allow

reconstructing the general tand-E0/q statistical relationships previously reported.

Trends for woods with different mechanical parameters merge into a single curve if anisotropic ratios (both elastic and of damping) are correlated between them, and with axial properties, as is indicated by the collected data. On the other hand, varying damping coefficient independently results in parallel curves, which coincide

I. Bre´maud (&)  J. Gril

Laboratoire de Me´canique et Ge´nie Civil, Universite´ Montpellier 2, CNRS, CC048, pl. E. Bataillon, 34095 Montpellier, Cedex 5, France

e-mail: iris_bremaud@hotmail.com I. Bre´maud

Laboratory of Forest Resources Circulatory System, Graduate School of Life and Environmental Sciences, Kyoto Prefectural University, Kyoto 606-8522, Japan B. Thibaut

with observations on chemically modified woods, either ‘‘artificially’’, or by natural extractives.

Introduction

Wood, with its cellular structure and its fibre-matrix composite cell-wall material, is

highly anisotropic. As a result, the actual orientation of ‘‘grain’’ (usually defined as

the orientation of all axial cellular elements) inside a wood piece will have strong

repercussions on its apparent mechanical properties (e.g. Bodig and Jayne 1982).

Although wood is most often used in longitudinal direction, most pieces of wood

actually include some grain angle (GA). Even in straight-grained trees, GA occurs

in sawn wood as a technical drawback of trunks not being truly cylindrical.

Moreover, grain is seldom perfectly straight inside trees, and more often than not

some grain deviations are present, either spiral grain which is common among

softwoods and some hardwoods, or more complex patterns such as interlocked grain

frequently found in tropical hardwoods, or wavy grain (e.g. Harris 1989).

In addition to GA dependence, the anisotropy of viscoelastic vibrational

properties, i.e. of specific dynamic Young’s (E0/q) and shear (G0/q) moduli and

damping coefficients (tand), also determines the vibration modes’ patterns of plates

(Caldersmith and Freeman 1990; Haines 2000), and the frequency response in

bending vibrations. The ratios between longitudinal and shear moduli and damping

coefficients are in good part responsible for the apparent frequency dependence

in the range of about 1–5–10 kHz, i.e. a diminution of E0/q and an augmentation of

tand, which also plays a role in the ‘‘timbre’’ for application in musical instruments

(Ono1996; Aizawa 1998; Haines 2000; Obataya et al. 2000).

The variability in anisotropy of vibrational properties will thus influence both

their GA dependence, and the ‘‘acoustical’’ response of wooden pieces. However,

this variability is not very well known, and information is much scattered. Amongst

possible sources of variation, the mean microfibril angle is recognised as the main

factor affecting both axial E0/q and tand, and their axial-to-shear anisotropy

(Norimoto et al. 1986; Obataya et al. 2000). The effect of microfibril angle on tand

and E0/q results in this two properties being correlated, and the relation is similar to

the case of GA effect (Ono and Norimoto 1983,1984,1985). The effect of cellular

organisation on damping coefficients is not clear, as the above relationships are

similar for softwoods and hardwoods with either diffuse- or ring-porous structure.

Yet, radial tand depends on the percentage of rays (Yano and Yamada 1985), and

the ratio between tangential and radial tand diminishes with increasing density,

indicating some effect of porosity (Aoki and Yamada 1972). On the other hand,

vibrational properties are very sensitive to chemical variations and/or modifications

within the cell wall, which can also modulate their anisotropy (Obataya et al. 2000).

Some extractives naturally present in wood can modify damping by as much as a

factor of 2 and can have smaller but anisotropic effects on moduli (Bre´maud et al.

2010b; Minato et al. 2010; Yano et al. 1995).

As apparent behaviour will depend on both intrinsic wood properties and

much less attempt has been made to describe viscoelastic dynamic properties, than

static, elastic ones. Schniewind and Barrett (1972) showed that creep at different

GA could be described by standard transformation formula and concluded that wood could be considered as a linear orthotropic viscoelastic material. Such formulas were also applied to GA dependence of dynamic modulus, but that of tand was approached either by statistical means or simplified formulas (Ishihara et al.

1978; Ono1983; Tonosaki et al.1983; Yano et al.1990).

This paper aims at gathering together the theory of GA dependence of vibrational properties, and their range of variability on different woods. GA dependence is described by a model based on transformation formula applied to complex compliances, and literature data on vibrational anisotropy are reviewed. This serves to predict the response of typical and contrasted wood types, and to interpret the relationship between damping coefficient and specific modulus.

Effect of grain angle on rigidity and damping: theory

The anisotropic organisation of wood in a trunk can be described by two systems of

axis (Fig.1): a global one aligned to the stem of the tree (and to ‘‘axial’’ samples

taken from it), which we call [R, T, L]; and a local one fitting the grain’s orientation, here noted [1, 2, 3]. When the trunk is closely cylindrical and no source of fibre deviation is considered, it can be assumed that both systems of axis, [1, 2, 3] and [R, T, L], coincide. However, in a trunk exhibiting grain deviation (spiralled or interlocked), or in pieces of wood sawn out of grain, directions 2 and 3 form an angle—from a few degrees to up to 45 in extreme cases—to the directions T and L.

Fig. 1 Schematic view of the systems of axis related to the trunk = ‘‘global’’ [R, T, L] and related to the

grain direction = ‘‘local’’ [1, 2, 3]. a In the particular case of interlocked grain within a trunk; b within a quarter-cut piece of wood cut along the stem axis: either a narrow (1–3 cm wide) board with interlocked grain, or a wide ([10–20 cm) plank with spiral grain

Mechanical properties in the global system of axis can be calculated from those in the local system, as a function of the angle (h) of grain with respect to the stem axis, by applying the transformation formula for elastic solids (e.g. Bodig and Jayne

1982). This gives, for the compliance along the stem axis:

SLL¼ S33cos4hþ Sð 44þ 2S32Þ cos2h sin2hþ S22sin4h

¼S33þ Sð 44þ 2S32Þ u þ S22u

2

1þ u

ð Þ2

ð1Þ

where u = tan2h and the component Sij of the compliance matrix represent the

strain response in direction i to stress applied in direction j.

In the case of linear viscoelastic solids, Eq.1can also be applied to the complex

compliance, with each component of the compliance matrix written in the form

S_ij¼ S0

ij iS00ij; where S0ij is the storage compliance and S00ij the loss compliance.

Separating the real and imaginary part of the expression leads to:

S0_LL¼S 0 33þ S044þ 2S032   uþ S0 22u 2 1þ u ð Þ2 ; S 00 LL¼ S00₃₃þ S00 44þ 2S0032   uþ S00 22u 2 1þ u ð Þ2 ð2Þ

Each compliance coefficient Sij*can be expressed by its inverse Qij*:

Q_ij¼ 1 S_ij¼ 1 S0_ij i S00 ij ¼S 0 ijþ i S 00 ij S02_ij þ S002 ij ) Q0_ij¼ S 0 ij S02_ij þ S002 ij ¼ 1 1þ tan d2 ij 1 S0_ij Q00_ij¼ S 00 ij S02 ij þ S002ij ¼ 1 1þ tan d2ij S00_ij S02 ij 8 > > > > < > > > > : ð3Þ where tan dij¼ S00_ij S0_ij¼ Q00_ij Q0_ij ð4Þ

are the damping coefficients, typically of the order of 1% in the time–temperature

domain considered here for wood, so that (tandij)2 1 and:

S0_ij Sij¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi S02_ij þ S002 ij q ; Q0_ij 1 S0_ij 1 Sij ; Q00_ijS 00 ij S02 ij tan dij Sij ð5Þ

The Qijcan be expressed using engineering notations:

QLL¼ EL; Q33 ¼ E3; Q22¼ E2; Q44 ¼ G32; Q32¼ E3=m32 ð6Þ

where EL, E3and E2are the Young’s moduli along the axial direction of the tree,

along the grain and across the grain (orthogonal to the radial direction),

respectively, G32is the shear modulus in the tangential plane and m32the Poisson’s

ratio relating the strain along 2 to the strain along 3 in the case of uniaxial loading along 3.

Then, the evolution of ‘‘apparent’’, global-scale storage modulus E0L

0

(h) and loss

modulus EL

00

(h) can be derived as a function of grain angle and properties at local scale:

E0_LðhÞ  E3 1þ u ð Þ2 1þ a0_{uþ b}0_u2; E 00 LðhÞ  E3tan d33 1þ u ð Þ2_ð1þ a00_{uþ b}00_u2_Þ 1þ a0_{uþ b}0_u2 ð Þ2 ð7Þ

In order to make the reading easier, dimensionless terms a0, a00, b0 and b00 are

used: a0¼S 0 44þ 2S 0 32 S0₃₃ b 0_¼S022 S0₃₃ a00¼S 00 44þ 2S 00 32 S00₃₃ b 00_¼S0022 S00₃₃ ð8Þ

From Eqs.5 and6:

a0 E3=G32 2m32 b0 E3=E2 a00E3=G32tan d44 2m32tan d32 tan d33 b00 Eð 3=E2Þ tan d22 tan d33 ð9Þ

In the case of wood, E3/G32is typically one order of magnitude higher than m32,

so that in the expression of a00 it will be convenient to isolate the contribution of

shear damping.

a00ðE3=G32 2m32Þ tan d44þ 2m32ðtan d44 tan d32Þ

tan d33

ð10Þ so that:

a00 a0tan d44

tan d33

ð1 þ qÞ where q¼ 2tan d44 tan d32

tan d44 m32 a0; b 00_{ b}0tan d22 tan d33 ð11Þ

It is convenient to use the specific storage modulus (E0/q) and specific loss

modulus (E00/q), i.e. the moduli divided by specific gravity (q), given that, on one

hand the specific Young’s modulus corresponds to the actual measurements by vibrational methods, on the other hand they are representative of the properties of

the cell walls (Norimoto et al.1986; Obataya et al.2000).

From these calculations of loss and storage moduli, the grain angle dependence of their ratio, the loss (or damping) coefficient tand can be obtained.

tan dLðhÞ¼ E00_LðhÞ E0_LðhÞ¼ E00_LðhÞ=q   E0_LðhÞ=q   ð12Þ

In the case of a beam made of wood layers with varying GA, it is possible to evaluate the global modulus of the beam by the application of the laminate theory, separately to the storage and loss moduli of each layer. In another paper by the authors

(Bre´maud et al.2010b), this has been applied to a configuration with interlocked grain.

Anisotropy of moduli and loss coefficients of woods

In order to overcome the fact that data are relatively scarce, or in any case much scattered, concerning the anisotropy of dynamic moduli and of damping coefficients,

Ta ble 1 Review of data on anisotrop ic ratio s of storag e mo duli and dampin g coeffici ents for diffe rent woods Materi al [Data sources] Speci fic gravity q E 0/qL (GPa) tan dL (% )T R = E 0 L G 0 ðTR ÞL E 0 L E 0 ðTR Þ G 0 ðTR ÞL E 0 ðTR Þ tan dG ðTR ÞL tan dL tan dðTR Þ tan dL tan dðTR Þ tan dG ðTR ÞL Standard ha rdwood a 0.65 22 T 14.8 14.0 0.9 R 11.4 8.0 0.7 Standard sof twood a 0.45 29 T 17.6 20.6 1.2 R 15.2 13.1 0.9 40 hardwoo ds a, b 0.10 –1.28 11–3 1 T 7.6 – 26.3 4.7 – 37.0 0.5 – 1.8 0.58 22.6 15.5 17.1 1.11 R 6.820.3 4.1 – 15.2 0.4 – 1.3 11.6 8.6 0.75 23 softwo ods a, b 0.26 –0.59 14–3 8 T 6.9 – 26.5 9.7 – 30.7 0.68 – 2.49 0.42 26.8 16.6 19.2 1.30 R 6.3 – 26.3 4.7 – 20.5 0.48 – 1.13 15.6 12.5 0.80 75 hardwoo ds c,d 0.09 –1.10 11–2 8 4.1– 14.8 T 5.3 – 24.5 1.29 – 3.91 0.64 20.5 7.1 13.5 2.21 25 softwo ods c, e 0.28 –0.61 12–3 2 4.3– 11.5 T 6.6 – 24.4 1.48 – 3.02 0.44 22 6.7 12.4 2.09 Picea sitchensis and P. spp. c, e 0.38 –0.49 27–3 1 6.0– 7.5 T 13.5 – 17.4 1.81 – 2.60 0.43 30.5 7.0 16.6 2.05 9 hardwoo ds f, g 0.08 –0.78 9–30 5.8– 15.2 R 7.0 – 18.1 1.43 – 3.52 0.55 21 7.9 13.4 2.25 3 softwo ods f, g 0.39 –0.48 24–2 8 6.3– 6.9 R 13.2 – 16.0 2.65 – 3.30 0.44 25 6.5 14.4 2.72

Ta ble 1 continu ed Materi al [Data sources] Speci fic gravity q E 0/qL (GPa) tan dL (% )T R = E 0 L G 0 ðTR ÞL E 0 L E 0 ðTR Þ G 0 ðTR ÞL E 0 ðTR Þ tan dG ðTR ÞL tan dL tan dðTR Þ tan dL tan dðTR Þ tan dG ðTR ÞL Picea sitchensis f, g, h 0.38 –0.52 22–3 6 5.5– 7.5 R 8.0 – 23.5 1.5 – 3.7 0.45 31 6.5 19.5 2.6 5 hardwoo ds i 0.48 –0.72 19–2 5 6.0– 8.5 T 11.1 – 18.5 2.63 – 3.47 0.61 22 7.3 R 14.3 2.95 7.6 – 11.7* 6.4 – 8.9 2.41 – 2.95 9.3* 8.1 2.74 Picea abies and P. sitche nsis i, j 0.41 –0.45 28–3 2 7.0– 8.1 T 20.4 – 29.4 2.18 – 4.07 0.43 30 7.5 24.9 3.13 8.1 – 11.2* R 13.8 – 21.4 2.07 – 3.89 9.6* 17.6 2.98 39 hardwoo ds j, k, l, m, n, o, p 0.48 –1.10 11–2 9 4.7– 15.0 R 3.7 – 18.5 1.48 – 4.12 0.66 18 9.4 7.4 2.47 14 softwo ods j, k, l, m, n, o, p, q 0.33 –0.68 16–3 7 3.5– 11.2 R 6.5 – 25.0 1.78 – 4.19 0.45 28 7.0 14.1 2.74 Picea abies and P. sitche nsis j,k,l,m,n,o,p, q 0.36 –0.56 24–3 6 5.1– 8.6 R 10.1 – 25.0 2.31 – 4.03 0.46 30 7.2 16.0 2.92 7 hardwoo ds l, n 0.56 –0.75 16–2 5.5 4.7– 10.9 R 5.8 – 15.2 6.8 – 11.6 0.76 – 1.30 1.33 – 2.38 1.83 – 3.25 1.13 – 1.48 0.62 20 7.4 8.8 8.0 0.95 1.85 2.51 1.36 5 softwo ods l, n 0.33 –0.54 17–3 2 5.6– 7.7 R 5.3 – 17.5 8.8 – 20.1 0.85 – 2.21 1.43 – 2.29 2.30 – 3.57 1.16 – 1.72 0.43 25 6.7 10.9 12.6 1.23 2.00 2.68 1.35

Ta ble 1 continu ed Materi al [Data sources] Speci fic gravity q E 0/qL (GPa) tan dL (% )T R = E 0 L G 0 ðTR ÞL E 0 L E 0 ðTR Þ G 0 ðTR ÞL E 0 ðTR Þ tan dG ðTR ÞL tan dL tan dðTR Þ tan dL tan dðTR Þ tan dG ðTR ÞL Picea abies & P. sitchensi s l, n 0.43 –0.54 24–3 2 6.7– 7.3 R 9.3 – 17.5 9.9 – 20.1 0.85 – 1.29 1.89 – 2.29 2.31 – 2.91 1.16 – 1.32 0.47 28 6.9 12.6 13.1 1.05 2.11 2.66 1.26 a (Guit ard and El Amri 1987 ) b (Green et al. 1999 ) c (Aiza wa 1998 ; A izawa et al. 1998 ) d (Bre ´maud et al. 2010 b ) e (Oka no 1991 ) f(Oba taya 1999 ) g (Ono 1980 ) h (Obataya et al. 2000 ) i (Ono and Nor imoto 1985 ) j(Barducci and Pasq ualini 1948 ) k (Bucur 2006 ) l(Calder smith and Free man 1990 ) m (Haines 2000 ) n (Ono 1996 ) o (Yano et al. 1992 ) p (Yano et al. 1995 ) q (Obataya et al. 2001 ) Sub scripts (TR) de signate ‘‘tr ansverse’ ’: the actual direc tion R or T is ind icated in the 5th column Num bers in itali cs are ranges of variation; those in bold are med ian values (or mean for small datasets) Co mpiled data had been obt ained in compar able hygro thermal condi tions (20–2 5 C and 60 ± 5%RH ) and frequ ency ran ges (50–2 ,000 H z, apar t fo r refere nce i above 4 kHz, for which original tan d3 valu es are noted by *)

the orders of magnitude for these anisotropic ratios are summarised in Table1, based on different samples and references. It can reasonably be assumed that values found in the literature had been obtained on straight-grained specimens where the global-scale system of axis [R, T, L] and the fibre-related one [1, 2, 3] can be superimposed. Axial-to-shear anisotropy

For storage moduli, mean anisotropic ratios are proposed by Guitard and El Amri

(1987) for ‘‘standard hardwoods’’ and ‘‘standard softwoods’’ (1st and 2nd row of

Table1, with actual ranges of variation shown in the 3rd and 4th row). Regarding

the differences between radial and tangential planes, EL/GTL and EL/GRL are

strongly correlated (R2 of 0.76 for both hardwoods and softwoods). The ratio

between tangential-longitudinal and radial-longitudinal shear moduli (GTL/GRL)

ranges from 0.50 to 0.98 (median 0.74) for hardwoods, and from 0.72 to 1.33

(median 0.95) for softwoods (Guitard and El Amri1987; Green et al. 1999).

Data on the axial-to-shear anisotropy of damping coefficients originate mostly from experiments both in flexural and in torsional vibration. They cover a wider

range of species for the axial-tangential plane (tandGTL/tandLfor 75 hardwood and

25 softwood species, 5th, 6th and 7th rows of Table1) than for the axial-radial one

(rows 8, 9 and 10 in Table1). Data on both tandGRL and tandGLT obtained on a

single sampling are not known; however, comparison on the same species across

different studies suggests that they are not very different (average tandGLT/tandGRL

of 1.04 on 6 hardwoods, of 0.95 on 5 softwoods).

Axial-to-shear anisotropic ratios are very weakly (for storage moduli) or not (for

tand) related to specific gravity (Table2 for data in the L–T plane, similar

observations can be made for the L–R plane). But they are strongly related to mechanical properties along the grain, which denotes that axial-to-shear anisotropy is primarily determined by cell-wall properties, notably by the mean microfibril angle

(Obataya et al.2000). However, cellular structure can also have an effect, as tandGRL

depends on the percentage of rays for hardwoods (Yano and Yamada1985). The

axial-to-shear anisotropy ratio in damping coefficients is strongly correlated to that in

storage moduli, as reported by Aizawa et al. (1998), Obataya (1999) and Obataya

et al. (2000). The relationship between tandGTL/tandLand E0L/G0TLis nearly the same

for softwoods and hardwoods, although there is more dispersion for the latter.

Table 2 Pearson’s correlation

coefficients between specific gravity (q), axial properties, and axial-to-shear (L–T plane) anisotropy ratios

Upper diagonal: softwoods, lower diagonal: hardwoods. Based on the data from

referencesa,b,c,e,m_{in Table1}

Hardw o ods N=157 ρ 0.27 0.26 0.36 -0.20 -0.18 N=154 Soft w oods -0.13 E’L/ρ 0.75 -0.45 0.58 0.79 -0.34 0.68 ' ' L TL E G -0.44 0.74 0.96 N=81 0.02 -0.65 -0.64 tanδL -0.63 -0.51 N=39 0.18 0.54 0.58 -0.72 tan tan GTL L δ δ 0.88 -0.06 0.69 0.88 -0.72 0.88 a"

Such correlations between the anisotropy in moduli and that in damping should

be taken into account for further analysis. Thus, in order to get a general view, mean

values of all parameters will be used, but for woods with very different elastic

anisotropy (correlated to E0L/q values), loss anisotropic ratios will have to be

adjusted, for example for ‘‘resonance’’ spruce wood (mostly Picea abies and

P. sitchensis), which is highly anisotropic (Obataya et al. 2000).

Axial-to-transverse anisotropy

As for shear, average axial-to-transverse (radial and tangential) anisotropic ratios of

elastic (storage) moduli were proposed for ‘‘standard’’ hardwoods and softwoods

(1st and 2nd rows of Table 1 with ranges of variations in 3rd and 4th rows)

according to Guitard and El Amri (1987) and Green et al. (1999). Anisotropy covers

a wide range of hardwoods, from the lowest values of ‘‘curly maple’’ (due to

systematic grain deviation and radial reinforcement by rays, Bucur 2006) to the

highest ratios for balsa. In the case of hardwoods, there is a strong correlation

(R = 0.90) between E0L/E0R and E0L/E0T and both ratios diminish with increasing

specific gravity, whereas for softwoods they are nearly independent of density and

more weakly linked between them.

For transverse damping coefficients, the majority of existing data concerns the

radial direction, while there is little information on the tangential one. However,

tandT is quite strongly correlated to tandR (R = 0.87 on 14 hardwoods), which may

be useful to get approximations of tandT. The tandT/tandR ratio diminishes with

increasing specific gravity, and ranges from 1.01 to 1.36 (mean 1.14) on 14

hardwoods. This range is comparable to the one on 4 softwoods, and the mean

tandT/tandR is about 1.05 for spruces. These relations are based on a moderate

number of species, but are consistent over three different studies (Aoki and Yamada

1972; Barducci and Pasqualini 1948; Ono and Norimoto 1985) and they sound

realistic, as the corresponding ratio E0R/E0T is comparable with the average on more

numerous species: 1.66 and 1.51 for hardwoods and softwoods, respectively.

Actual data on axial-to-tangential anisotropy of tand are listed in Table 1 (11th

and 12th rows for 5 hardwoods and 2 spruces). The axial-to-radial ratio (tandR/

tandL) ranges from 1.49 to 4.12 (median 2.47) over 39 hardwoods species and from

1.78 to 4.19 (median 2.74) over 14 softwood species (Table 1 from row 11 to the

end).

The axial-radial anisotropy in damping coefficient is correlated to that in storage

moduli and to axial properties (Table 3). However, most relations are less strong

than in the case of shear anisotropy, which may denote a bigger importance of

cellular organisation when compared to cell-wall properties.

Poisson’s ratio

Elastic/storage Poisson’s ratio t0_LT ranges from 0.31 to 0.73 on hardwoods, and from

0.36 to 0.6 on softwoods, with respective ‘‘standard’’ values of 0.46 and 0.43

Experimental values of Poisson’s loss factor for wood in a time–temperature domain close to audible frequencies and c.20C are not known. For isotropic materials, it would be about one decade smaller than shear-loss coefficient (Pritz

2007), but this is not necessarily true for anisotropic cellular materials. According to

Eq.11, the term q would tend towards zero if tand32 (Poisson’s loss factor) was

close to tand44 (in shear). If tand32 was negligible compared to tand44, the term

q would contribute about up to 10% to the term a00used for calculations.

Variability and relative influence of axial-to-shear and axial-to-transverse parameters

Over all the collected data on both hardwoods and softwoods species, the

axial-to-shear anisotropy ranged from 4 to 30 (mean 14) for E03/G023 and from 1.3 to 3.9

(mean 2.25) for tand44/tand33(from now on, it is admitted that values that are listed

in Table1 in the [R, T, L] system of axis are equivalent to those in the [1, 2, 3]

system of axis). The axial-to-tangential anisotropy ranged from 5 to 40 (mean 18)

for E03/E02and from 1.6 to 4.4 (mean 2.9) for tand22/tand33 (based on both actual

tangential data and estimates from radial ones). Ranges for hardwoods and

softwoods are resumed in Table4.

For moderate GA (up to 20–25), the total variability in shear anisotropy accounted for most of the differences in GA dependence of storage modulus

(Fig.2a), while the full variability in tangential anisotropy also contributed

significantly to damping coefficient (Fig.2b). Over the whole range of GA, L/GTL

anisotropy still contributed the most important (quantitatively) to the variability in

GA dependence for E0, while L/T anisotropy contributed at least as much to the GA

dependence of tand.

Grain angle dependence of vibrational properties for different types of woods

In order to illustrate (Fig.3) the grain angle dependence of dynamic mechanical

properties for different wood types, average values for hardwoods and softwoods

were defined (Table4). Mechanical parameters are also summarised for the

highly anisotropic ‘‘resonance’’ spruces and for a species (African padauk) with

Table 3 Pearson’s correlation

coefficients between specific gravity, axial properties, and axial-to-radial anisotropy ratios

Upper diagonal: softwoods, lower diagonal: hardwoods. Based on data from references

a,b,j,k,l,m,n,o,p,q in Table1 Hardw o ods N=123 ρ 0.19 0.21 0.09 0.28 0.20 N=120 Soft w oods -0.28 E’L/ρ 0.65 -0.63 0.31 0.50 -0.46 0.60 '' L R E E -0.23 0.48 0.95 N=53 -0.28 -0.65 -0.40 tanδL -0.33 -0.26 N=64 0.10 0.71 0.64 -0.69 tan tan R L δ δ 0.70 0.02 0.79 0.92 -0.52 0.85 b"R

Ta ble 4 Summ ary of dynami c mech anical propert ies and their anisotrop ic ratios fo r docume nted rep resentat ive hardwoo ds and softwo ods and two atyp ical w oods Materi al q E 0/q3 (GPa) E 00 3 /q (MPa) tan d3 (% ) E 0 3 G 0 23 E 0 3 E 0 2 tan dG 23 tan d3 tan d2 tan d3 E 0 3 G 0 13 E 0 3 E 0 1 tan dG 13 tan d3 tan d1 tan d3 H ardwood s Media n 0.66 20.5 163 8.1 13.7 15.4 2.27 2.92 (*) 11.7 7.8 2.14 2.68 Rang e 0.08 – 1.28 9 – 32 82 – 339 4.1 – 15.2 6 – 24 5 – 27 1.3 – 3.9 6 – 20 4 – 17 1.3 – 3.5 1.5 – 4.1 Soft woods Media n 0.44 25.1 180 7.1 14.1 18.5 2.14 3.08 (*) 14.4 11.6 2.16 2.90 Rang e 0.26 – 0.61 13 – 36 77 – 311 4.3 – 14.8 7 – 29 10 – 32 1.5 – 3.0 6 – 26 8 – 25 1.4 – 3.3 1.8 – 4.2 ‘‘Resonance’ ’ Spru ce Media n 0.45 30.0 199 6.8 16.8 25.9 2.32 3.36 (*) 15.6 14.7 2.22 2.92 Rang e 0.35 – 0.56 26 – 36 177 – 292 5.1 – 8.3 12 – 28 23 – 31 1.8 – 3.7 9 – 23 12 – 25 1.9 – 3.7 2.31 – 4.0 A frican pada uk (Pter ocarp us soyau xii ) Mean 0.75 17.3 86 4.8 9.7 11.5 ( ) 2.13 3.5 ( ) 9 ( ) 7 ( ) 2.1 ( ) 3.1 ( ) Da ta fo r all hardwoo ds, softwo ods and ‘‘reso nance spruce’ ’ summ arised fro m refere nces reviewe d in Tabl e 1 * Mean of experime ntal data on tang ential plane and of esti mations fro m da ta in rad ial plan e s V alues adjusted by com paring two spe cies of the same genus with simi lar axial propert ies

abnormally low damping coefficient and reduced anisotropy (Bre´maud et al.

2010b).

The relationship between elastic anisotropy and specific Young’s modulus along

the grain is clearly visible in the grain angle dependence of E0/q for different wood

types (Fig.3a): for a small GA of 5, E0_{/q decreases of 11% (when compared to 0}

GA) for spruce, but only of 5% for padauk. This has non-negligible practical repercussions, as GA of a few degrees are common in sawn pieces of wood; 5–10 can quite easily occur in softwood planks, and about 10–15 are not uncommon in hardwoods with grain deviations. For a GA of only 7, ‘‘resonance’’ qualities of spruce become equivalent to the average softwoods along the grain. The GA dependence curves of spruce and mean softwoods join up for an angle C15, those of mean hardwoods and of padauk for GA C 10. For angles bigger than 20, the tendencies are comparable for all wood types represented.

The scheme of GA dependence is quite different for specific loss modulus E00/q

which remains quite stable for moderate GA of up to 10–15, or even slightly

increases (Fig.3b). This denotes the fact that, over small angles, the damping

coefficient increases a bit faster than Young’s modulus decreases. Wood types

compare differently than in the case of E0/q: GA dependence of mean hardwoods

and softwoods is very similar, and ‘‘resonance’’ spruce is less different in terms of specific loss modulus, while padauk wood is here the most atypical one. The latter is also noticeable for its systematically lower damping coefficient tand over varying

GA (Fig.3c). The damping coefficient of mean hardwoods and softwoods increases

according to grain angle in a nearly parallel way, and the trend for ‘‘resonance’’ spruce starts to be dissociated from that for mean softwoods for GA C 10.

The calculated trends in GA dependence for tand and E0/q result in relationships

between these properties (Fig.3d) quite similar to those reported by Ono and

Norimoto (1983,1984,1985); this will be discussed below. Such relationships are

Fig. 2 Relative influence of the anisotropy ratios (longitudinal to shear, and longitudinal to tangential)

on the grain-angle dependence of (a) dynamic Young’s modulus and (b) loss factor. Plain curves: full range of variation in axial-to-shear anisotropy ratios with axial–tangential ones fixed at their mean values;

Fig. 3 Dependence of dynamic mechanical properties on grain angle (L–T plane) calculated using the

sets of parameters from Table 4for different types of woods. Trends in specific storage modulus (a),

specific loss modulus (b) and damping coefficient (c) at the global scale, as a function of grain angle.

d Calculated evolution of tand plotted against that of E0_/q

very comparable for mean hardwoods and softwoods and for ‘‘resonance’’ spruce,

while that for padauk systematically remains ‘‘shifted’’ downwards low tand values.

Calculations of GA dependences of vibrational properties are in quite good

agreement with published experimental values, as illustrated in Fig. 4 for two

contrasted wood types: (1) ‘‘resonance’’ spruce with a high E03/q, cut every 10 or

15 in L–R plane (Ono 1983; Tonosaki et al. 1983). (2) Rio rosewood (Dalbergia

nigra) cut every 10 in L–TR plane (Yano et al. 1995) is compared with calculations

based on padauk parameters from Table 4, as both these medium-heavy

Legumi-nosae species have a reduced anisotropy, and abnormally low damping coefficients

(due to their particular extractives). The calculations of the authors (using Eqs. 7

and12) provide good predictions for both ‘‘extreme’’ types of woods, including loss

parameters which GA dependence had been hitherto approached either by statistical

Relationship between damping coefficient and specific storage modulus along with varying grain angle

In Fig.3d, it can be noted that calculated GA dependence trends for tand and E0/q

are related in a way very similar to the relationships proposed by Ono and Norimoto

(1983, 1984, 1985). Calculated relations are compared to these statistical

regressions (in the form of tand = A9[E0/q]-B) in Fig.5, where curves (a) and

(b) are observed as a function of grain angle, and curves (c) and (d) on wood along the grain, i.e. as a function of microfibril angle. These four regression curves are rather close to each other, suggesting that microfibril- and grain-angle have similar consequences. The calculated curves of this study (for mean hardwoods and softwoods and spruce) are also quite close to each other, and they are in quite good agreement with the regressions over the extreme ranges of GA and of properties, but they are of a less concave form. This is partly because of the concavity of a power curve fitting in itself, but variations in mechanical properties and anisotropic ratios

should also modulate the shape of the tand-E0/q relation.

If all anisotropic parameters (axial-shear and axial-transverse, for storage moduli and loss coefficients) are made to vary from their minimum to maximum values,

while keeping axial properties constant, the tand-E0/q relationship increases slower

or faster and of course maximum tand values are different, yet the shape of the

relation remains comparable (Fig.6a). Moreover, as introduced above, anisotropic

ratios are correlated to axial mechanical properties (Tables2 and 3). If axial

properties are roughly adjusted to anisotropic ratios, calculations for low- and

high-anisotropy woods join up into a common trend (Fig.6b).

This latter analysis implied that all anisotropic ratios tend to increase or decrease together. However, there is little experimental data (obtained on the same samples) concerning the relations between shear- and transverse-anisotropy of damping

coefficients. tand44/tand33 should be primarily affected by cell-wall properties

(Aizawa et al.1998; Obataya et al.2000) while tand22/tand33could presumably be

Fig. 4 Comparison of calculated (using Eqs.7 and 12) and experimentally determined grain-angle

dependence for contrasted types of wood. Curves: calculated with L–R anisotropic parameters for spruce

(axial E0/q adjusted to 32GPa) and padauk (Table4). Triangles: Sitka spruce cut at angles in L–R plane

(Ono1983; Tonosaki et al.1983). Circles: Rio rosewood (Dalbergia nigra) cut at angles in L–TR plane

Fig. 5 Comparison of the trends in the damping coefficient—specific modulus relationship: calculated as

a function of GA with Eqs.7–12 and mean anisotropic coefficients (listed in Table4, symbols: see

Fig.3), and regression curves (thick patterned lines a, b, c, d) from the literature. (a): 5 hardwoods tested

along the 3 axes R, T, L at frequencies of 4–20 kHz (Ono and Norimoto1985); (b): Sitka spruce with

varying GA in L–R plane (Ono and Norimoto1983); (c) 25 softwood species along the grain (Ono and

Norimoto1983); (d) 30 hardwood species along the grain (Ono and Norimoto1984)

Fig. 6 Variations in the calculated tand–E0_{/q (0–90 in L–T plane) curve, when varying all anisotropic}

ratios from their maximum to their minimum. Storage and loss anisotropic ratios are considered as

correlated. (a) Properties along the grain are kept constant; (b) E0

3/q and tand3 are adjusted, i.e. they are

considered as correlated to anisotropic ratios

more affected by cellular organisation. Varying them separately (Fig. 7) describes

as much or even more variability in the tand-E0/q relation as total variations in

anisotropy do. If tand₄₄ comes close to tand22, the shape of the relation tends

towards a linear one. On the contrary, the smaller tand44/tand22, the more concave

the tand-E0/q curve. Altogether, variations in damping anisotropic ratios define a

Finally, if anisotropic parameters are kept constant, and only the axial specific modulus varies, a deviation (from the curve calculated with all mean parameters) is

observed over small GA, but trends merge again for increasing grain angles (Fig.8).

Fig. 7 Variations in the calculated tand–E0/q (0–90 in L–T plane) curve, when varying loss anisotropic

ratios (axial-to-shear and axial-tangential) from their maximum to their minimum, while keeping elastic anisotropic ratios and axial properties at their mean values

Fig. 8 Variations in the calculated tand–E0_{/q (0–90 in L–T plane) curve, when varying individually}

While variations in axial damping coefficient alone, all else being equal, result in

tand-E0/q curves that remain shifted towards lower- or higher-tand values, over the

whole range of grain angle. This can describe the case of chemical modifications affecting mostly tand, such as for some extractives: parallel but shifted curves are observed either between different species, as was illustrated above for padauk and rosewood, or between sapwood and heartwood of the same species (Bre´maud et al.

2010a; Matsunaga et al.1996; Yano1994; Yano et al.1990). Analyses on this topic should however be a bit more detailed, as chemical modifications can also have a

smaller but non-negligible effect on E0/q and elastic anisotropy (Minato et al.2010;

Obataya et al.2000; Yano et al.1995).

Conclusion

This article aimed at evaluating the variability in the anisotropy of wood vibrational properties, and its consequences on their grain angle dependence. The combination of a theoretical mechanical analysis and a review of literature data on anisotropy of dynamic properties has allowed to summarise the following points:

• Axial-to-shear anisotropy in damping (or loss) coefficients (tand) ranges from

1.3 to 3.9 for hardwoods and from 1.4 to 3.3 for softwoods. Axial-to-transverse ratios range from 1.5 to 4.4 and from 2.2 to 4.4, respectively.

• The anisotropy in tand is correlated to that in specific elastic moduli. Anisotropic

ratios are also positively correlated to axial E0/q.

• Calculations based on transformation formula applied to complex compliances

are in good agreement with published experimental data. They efficiently represent the GA dependence both for storage modulus and loss properties, and allow predicting the practical repercussions of GA occurrence in pieces of different types of wood.

• Calculated trends in tand and in E0/q allow reconstructing the general relation

between these two properties that had been previously reported by statistical means. However, they lead to a less concave curve than the reported power fit.

The more or less concave shape of this tand-E0/q relation mainly depends on the

ratio between shear and transverse damping coefficients.

• The tand-E0/q relation for woods with different mechanical parameters is

distributed along a unique curve if all anisotropic ratios evolve in conjunction

together and with axial E0/q and tand. On the contrary, variations in tand33

independently of E03/q result in parallel curves whatever the orientation be,

which coincide with experimental observations on chemically modified woods (artificially or by natural extractives).

The theoretical GA dependence of vibrational properties that was applied here in

the simpler case of constant, homogeneous GA, could also serve as the basis for

studying more complex phenomena, such as the repercussions of interlocked or

wavy grain. In addition, compiled data on anisotropy of dynamic mechanical

properties could also be useful for other types of analysis, such as for modelling the

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